The co-axial system of circles given by ${x^2} + {y^2} + 2gx + c = 0$ for $c < 0$ represents

The equation of the circle whose radius is $3$ and which touches the circle $x^2+y^2-4x-6y-12=0$ internally at $(-1,-1)$ is

The equation of the circle which passes through the point $(2a, 0)$ and whose radical axis is $x = \frac{a}{2}$ with respect to the circle $x^2 + y^2 = a^2$ is:

The centre of the circle passing through the point $(1,1)$ and orthogonal to the circles $x^2+y^2+3x-5y+7=0$ and $x^2+y^2-6x-10y+9=0$ is

$A$ circle passes through $(0, 0)$ and $(1, 0)$ and touches the circle ${x^2} + {y^2} = 9$. Find the centre of the circle.

Suppose $A(2,3)$ and $B$ are the points of intersection of two circles. The points $P$ lying on one circle and $Q$ lying on the other circle are such that $BP$ and $BQ$ constitute the diameters of the circles. If the slopes of the radical axis and $PQ$ are $3/4$ and $a/b$ respectively,then the value of $3a+4b$ is